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Accurate Quantum Chemistry forLarge Molecules
Fred Manby
Centre for Computational Chemistry, School of Chemistry
University of Bristol
Outline
Three problems in quantum chemistry
Three solutions
Accurate quantum chemistry for large molecules
Quantum chemistry
HΨ = EΨ
(One possible) hierarchy of methods
HF −→ MP2 −→ CCSD −→ CCSD(T) −→ CCSDT −→ · · · −→ FCI
(One possible) hierarchy of basis sets
VDZ −→ VTZ −→ VQZ −→ V5Z −→ · · · −→ ∞
Chemical accuracy ∼ a few kJ mol−1
Accurate quantum chemistry for large molecules
Benchmarking standard methods
• Set of simple reactions (from Helgaker, Jørgensen and Olsen)
CO2 + 4H2 −→ CH4 + 2H2O N2 + 3H2 −→ 2NH3
C2H2 + H2 −→ C2H4 CO + H2 −→ CH2O
CH2O + 2H2 −→ CH4 + H2O F2 + H2 −→ 2HF
HCN + 3H2 −→ CH4 + NH3 O3 + 3H2 −→ 3H2O
C2H2 + 3H2 −→ 2CH4 CH2 + H2 −→ CH4
CO + 3H2 −→ CH4 + H2O 2CH2 −→ C2H4
HNO + 2H2 −→ H2O + NH3
• Plot normal distributions of errors (in kJ mol−1)
Accurate quantum chemistry for large molecules
Errors for benchmark reaction energies in kJ mol−1
CCSD(T)
−80 80 −80 80 −80 80 −80 80
CCSD
−80 80 −80 80 −80 80 −80 80
MP2
−80 80 −80 80 −80 80 −80 80
HF
−80 80 −80 80 −80 80 −80 80
VDZ VTZ VQZ V5Z
Accurate quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size
Steep scaling of effort wrt basis set size
Slow convergence wrt basis set size
Excited states
Quantum dynamics of nuclei
Solvation
Multireference methods
Accurate quantum chemistry for large molecules
Steep scaling of effort wrt molecular size
(H2O)n
2 3 4 5 6n
0
500
1000
1500tim
e/s
CCSD(T)CCSDMP2HF
Accurate quantum chemistry for large molecules
Steep scaling of effort wrt basis set size
H2O with MP2/cc-pVnZ
VDZ VTZ VQZ V5Z V6Zbasis set
0
50
100tim
e/s
Accurate quantum chemistry for large molecules
Slow convergence in quantum chemistry
error ∝ [basis size]−1
time ∝ [basis size]4
⇒ error ∝ time−1/4
time0
erro
r10 000-fold improvement in CPU speed gives only one order of magnitude
Accurate quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size
Steep scaling of effort wrt basis set size
Slow convergence wrt basis set size
Accurate quantum chemistry for large molecules
Canonical and local orbitals
Indinavir — a canonical orbital
Accurate quantum chemistry for large molecules
Canonical and local orbitals
Indinavir — a local orbital
Accurate quantum chemistry for large molecules
Rapid decay of correlation energy
0 5 10 15 20 25 30Interorbital Distance [bohr]
1e-06
0.0001
0.01
1
Incr
emen
tal
Cor
rela
tio
n E
nerg
y [
Har
tree
]
1 kcal/mol
Indinavir
rh
ob
72.
33
Accurate quantum chemistry for large molecules
Local correlation theories
• Use localized orbitals
• Take advantage of the short-ranged nature of correlation
• Pioneered by Pulay and Saebø
• Current implementation by Werner and Schutz
• Achieves linear scaling with respect to system size
Accurate quantum chemistry for large molecules
LMP2/VDZ on (Gly)n: Werner et al.
2 4 6 8 12 16 20n
0
50
100
150
CP
U-t
ime
/ min
ute
standard MP2local MP2
Accurate quantum chemistry for large molecules
Local coupled cluster (Schutz and Werner)
Glyn with LCCSD/VDZ and LCCSD(T)/VDZ
0 2 4 6 8 10 12 14 16n
0
5000
10000
15000
20000
25000
30000
35000
40000C
PU ti
me
/ s(T) CCSD Iteration
(~N )
L(T) (~ N)
(~N )7 6
LCCSD Iteration(~N)
Accurate quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size
Slow convergence wrt basis set size
Accurate quantum chemistry for large molecules
Density fitting
All ab initio quantum chemistry needs 2-electron integrals∫d~r1
∫d~r2 ψ
∗p(~r1)ψq(~r1)
1
r12ψ∗r(~r2)ψs(~r2)
|pq) |rs)Idea is to fit these orbital product densities in a set of functions
|pq) =∑A
DpqA |A)
Accurate quantum chemistry for large molecules
Density fitting
• First used by Boys and Shavitt for H3 (1959)
• Used by Baerends and Dunlap in density functional theory (1970s)
• Recently used in ab initio methods: Ahlrichs, Feyereisen, Komornicki
FRM, Knowles, Lloyd PRL 87 163001 (2001); JCP 115 9144 (2001)
Werner, FRM, Knowles, JCP 118 8149 (2003)
FRM, JCP 119 4607 (2003); Ten-no, FRM, JCP 119 5358 (2003)
Schutz, FRM, PCCP 5 3349 (2003)
Schutz, Werner, Lindh, FRM, JCP 121 737 (2004)
May, FRM, JCP 121 4479 (2004)
Accurate quantum chemistry for large molecules
Efficiency of density fitting
H2O with MP2/cc-pVnZ
VDZ VTZ VQZ V5Z V6Zbasis set
0
100
200
300
400
500
600
700tim
e/s MP2
DF-MP2
Accurate quantum chemistry for large molecules
Combining local and density fitting methods
• Use localized orbitals Werner, FRM, Knowles, JCP 118 8149 (2003)
• Perform density fitting
• Fit products of local orbitals in localized fitting expansions
|ia) ≈∑
A near i,a
DiaA |A)
Accurate quantum chemistry for large molecules
DF-LMP2 performance
Indinavir in cc-pVTZ (2008 bf)
CPU time/second
LMP2 DF-MP2 DF-LMP2
Integrals 25540 2992 2816
Transformation 56620 4795 970
Solve 0 3364 362
Assemble 0 82663 38
Iteration 3772 0 3775
Total MP2 86177 93914 8247
Werner, FRM, Knowles, JCP 118 8149 (2003)
Accurate quantum chemistry for large molecules
DF-LMP2/VDZ on (Gly)n
2 4 6 8 12 16 20n0
50
100
150
CP
U-ti
me
/ min
ute LMP2
DF-MP2DF-LMP2
Accurate quantum chemistry for large molecules
DF-LMP2 accuracy
Comparison of MP2 and DF-LMP2 reaction energies (kcal/mol)
VTZ VQZ
MP2 DF-LMP2 MP2 DF-LMP2
I −16.28 −16.29 −15.24 −15.24
II −51.50 −51.49 −50.83 −50.79
III −151.58 −151.58 −156.27 −156.27
I HF + 2-butene → 2-fluorobutane
II
III THF + 2 H2O2 → γ-butyrolactone + 3 H2O
Accurate quantum chemistry for large molecules
Other DF-local theories
• DF-HF Polly, Werner, FRM, Knowles, Mol. Phys. in press (2004)
◦ Needed since the DF-LMP2 program was so fast
• DF-LCCSD(T) Schutz and FRM, PCCP 5 3349 (2003)
◦ In progress, but indicates 100-fold improvement in speed over LCCSD(T)
• DF-LMP2 gradients Schutz, Werner, Lindh, FRM, JCP 121 737 (2004)
Accurate quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size — density fitting
Slow convergence wrt basis set size
Accurate quantum chemistry for large molecules
The origin of slow convergence
• H only has terms like 1/r12 and −12∇
2
• 1/r12 blows up when electrons coalesce
• HΨ(r12)/Ψ(r12) = E does not
• Cancellation of divergence must be from KE
• −12∇
2Ψ(r12) = − 1
r12for small r12
• −12∇
2r12 = − 1
r12
Accurate quantum chemistry for large molecules
Slow convergence
• Orbital expansions are not very good for describing correlation
• Orbitals expanded about nuclei, not about other electrons
• Simple solution: include terms that depend on r12 in the wavefunction
Accurate quantum chemistry for large molecules
Explicitly correlated theories
Hylleraas’ pioneering calculations on helium — 1929
0 100 200 300 400 500 600 700number of terms
-9
-8
-7
-6
-5
-4lo
g(er
ror)
orbital-based wavefunctionsHylleraas wavefunctions
Accurate quantum chemistry for large molecules
DF-MP2-R12 theory
• Based on MP2-R12 theory (Kutzelnigg, Klopper et al.)
• Uses density fitting FRM, JCP 119 4607 (2004)
• Uses an optimal correlation factor Andy May, FRM, JCP 121 4479 (2004)
• Local versions under development (FRM and H-J Werner)
Accurate quantum chemistry for large molecules
Benchmarking explicitly correlated theories
1 CO + SO3 → CO2 + SO2 2 C2H2 + H2O → CH3CHO
3 CO + Cl2 → COCl2 4 furan + H2S → thiophene + H2O
5 CO + CH3OH → HCOOCH3 6 CO + NH3 → HCONH2
7 CO + H2O → CO2 + H2 8 CS2 + 2 H2O → CO2 + 2 H2S
9 H2CCO + HCHO → C2H4O + CO 10 H2O2 + H2 → 2 H2O
11 C2H2 + H2 → C2H4 12 C2H4 + H2 → C2H6
13 HCHO + H2 → CH3OH 14 C2H6 + H2 → 2 CH4
15 CO + H2 → HCHO 16 CH4 + 4 H2O2 → CO2 + 6 H2O
17 NH3 + 4 H2O2 → HNO3 + 5 H2O 18 CO + H2O2 → CO2 + H2O
19 SO2 + H2O2 → SO3 + H2O 20 HNCO + NH3 → NH2CONH2
Accurate quantum chemistry for large molecules
Benchmarking explicitly correlated theories
0 2 4 6 8 10 12 14 16 18 20Reaction number
-2
-1
0
1
2
3
4E
nerg
y di
ffer
ence
(kca
l/mol
)
AVTZAVQZAVTZ+R12
CCSD(T) correlation energy contributions relative to AVQZ+R12
Accurate quantum chemistry for large molecules
Three problems in quantum chemistry
. . . and three solutions
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size — density fitting
Slow convergence wrt basis set size — explicit correlation
Accurate quantum chemistry for large molecules
Some applications
DNA intercalators:
J Platts (Cardiff)
PHBH enzyme: W Thiel (Mulheim),
R Mata, H-J Werner (Stuttgart)
Gas-phase conformation of enkephalins:
J Hirst (Nottingham)
Chorismate mutase:
Fred Claeyssens, AJM, JNH
Accurate quantum chemistry for large molecules
Chorismate mutase
• Protein bulk treated by molecular mechanics (QM/MM)
• Calculations by Fred Claeyssens
Accurate quantum chemistry for large molecules
Chorismate mutase
-2 -1 0 1 2reaction coordinate / angstrom
-30
-20
-10
0
10
20
30
40en
ergy
/ kc
al m
ol-1
HFMP2CCSD(T)
Accurate quantum chemistry for large molecules
Future work
• Method development
◦ Correlated alternatives to Hartree-Fock theory
◦ Optimal control, interaction of light and matter (with GGBK)
◦ Quantum mechanics of light nuclei
◦ Combined molecular mechanics and quantum mechanics (with JNH, AJM)
• Applications
◦ High-level quantum chemistry of enzymes (with AJM, JNH)
◦ Quantum chemistry at interfaces (with NLA, JPR)
Accurate quantum chemistry for large molecules
Conclusions
• Chemical accuracy can now be achieved for large molecules
• New methods combine three key developments
◦ Local description of correlation
◦ Density fitting
◦ Explicit correlation
• Methods available in Molpro quantum chemistry package
◦ Distributed to 300 research groups across the world
Accurate quantum chemistry for large molecules